Particle size as key frame
While i am sure you have a reason to use the older version of Blender (update is much recommended though).

That said, you should be able to get it done by doing the following:


Go to frame 1, where the scale is supposed to be 0.001
Enter the value of 0.001 and then with the mouse over the scale field, press I to set a keyframe for that attribute at frame 1.
Now jump to frame 300 and enter the new value (0.05) for the scale and again with the mouse over the scale field press I to set another keyframe.


If you now play the animation, the scale value will be interpolated between 0.001 and 0.05 over the run of 300 frames. If you want a linear interpolation, you need to go in the graph editor and under key menu -> Interpolation Mode -> Linear. Or just press T and select Linear, like in the picture below.


I packed all in a little visual demonstration:


Happy Blending